To solve the given problem, we need to subtract the repeating decimal [tex]\(7.\overline{2}\)[/tex] from the decimal [tex]\(2.8\)[/tex].
1. First, we express [tex]\(7.\overline{2}\)[/tex] as a fraction. Consider the repeating decimal:
[tex]\[
x = 7.2222 \ldots
\][/tex]
Multiplying both sides by 10 to shift the decimal point:
[tex]\[
10x = 72.2222 \ldots
\][/tex]
Subtracting the original [tex]\(x = 7.2222 \ldots\)[/tex] from this result:
[tex]\[
10x - x = 72.2222 \ldots - 7.2222 \ldots
\][/tex]
[tex]\[
9x = 65
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{65}{9}
\][/tex]
Therefore:
[tex]\[
7.\overline{2} = \frac{65}{9}
\][/tex]
2. Convert [tex]\(\frac{65}{9}\)[/tex] to its decimal form:
[tex]\[
\frac{65}{9} \approx 7.2222\overline{2}
\][/tex]
3. Next, perform the subtraction:
[tex]\[
2.8 - 7.2222\overline{2}
\][/tex]
Given the above computations:
[tex]\[
2.8 - 7.2222\overline{2} \approx -4.4222\overline{2}
\][/tex]
So, the answer to [tex]\(
2.8 - 7.\overline{2}
\)[/tex] is approximately [tex]\(-4.4222222222222225\)[/tex].
